With the rapid integration of renewable energy, wide-band oscillations caused by interactions between power electronic equipment and grids have emerged as one of the most critical stability issues. Existing methods are usually studied for local power systems with around one hundred nodes. However, for a large-scale power system with tens of thousands of nodes, the dimension of transfer function matrix or the order of characteristic equation is much higher. In this case, the existing methods such as eigenvalue analysis method and impedance-based method have difficulty in computation and are thus hard to utilize in practice. To fill this gap, this paper proposes a novel method named the smallest eigenvalues based logarithmic derivative (SELD) method. It obtains the dominant oscillation modes by the logarithmic derivative of the k-smallest eigenvalue curves of the sparse extended nodal admittance matrix (NAM). An oscillatory stability analysis tool is further developed based on this method. The effectiveness of the method and the tool is validated through a local power system as well as a large-scale power system.